In fact, '때' is not a particle but a noun which means a specific time or period.
It is right that '에' is a particle indicating a time or place.
So, actually '여름 때' is short for '여름 때에'. However, we usually don't use '에' together with '때'.
I don't know the exact reason because I am not a grammarian but I think it is because the sound of 'ㅐ' in '때' and 'ㅔ' in '에' is similar.
It is hard to say the difference between '여름에' and '여름 때' but it seems to me '여름 때' emphasizes the specific season or time period of '여름'.
'(으)면' can mean both 'if' or 'when', while 'ㄹ때' means 'when' only.
Please compare the following examples.
1) 내가 공부하면 그는 항상 잠을 잔다. When I study, he always sleeps.
내일 비가 오면 나는 밖에 나가지 않겠다. If it rains tomorrow, I will not go out.
2) 내가 노래할 때, 그는 귀를 막고 있었다. When I was singing, he was closing his ears.